New metrics are introduced in the space of random measures and are applied, with various modifications of the contraction method, to prove existence and uniqueness results for self-similar random fractal measures. We obtain exponential convergence, both in distribution and almost surely, of an iterative sequence of random measures (defined by means of the scaling operator) to a unique self-similar random measure. The assumptions are quite weak, and correspond to similar conditions in the deterministic case.

The fixed mass case is handled in a direct way based on regularity properties of the metrics and the properties of a natural probability space. Proving convergence in the random mass case needs additional tools, such as a specially adapted choice of the space of random measures and of the space of probability distributions on measures, the introduction of reweighted sequences of random measures and a comparison technique.

We study the transformation of a non-stationary point process ξ on ℝn into a weakly stationary point process ͂ξ, with ͂ξ(B) = ξ(Φ-1(B)), where B is a Borel set, via a deformation Φ of the space ℝn. When the second-order measure is regular, Φ is uniquely determined by the homogenization equations of the second-order measure. In contrast, the first-order homogenization transformation is not unique. Several examples of point processes and transformations are investigated with a particular interest to Poisson processes.

We discuss necessary and sufficient conditions for power-law and polynomial models to be correlation functions on bounded domains. These results date back to unpublished work by Matheron (1974) and generalize the findings of Gneiting (1999).

We consider a fully connected queueing network in which customers have one direct and many alternative routes through the network, and where customer routeing is dynamic. We obtain an asymptotically optimal routeing policy, taking the limit as the number of queues of the network increases. We observe that good policies route customers directly, unless there is a danger of servers becoming idle, in which case customers should be routed alternatively so as to avoid such idleness, and this leads to policies that perform well in moderate-sized networks.

Haploid population models with non-overlapping generations and fixed population size N are considered. It is assumed that the family sizes ν1,…,νN within a generation are exchangeable random variables. Rates of convergence for the finite-dimensional distributions of a properly time-scaled ancestral coalescent process are established and expressed in terms of the transition probabilities of the ancestral process, i.e., in terms of the joint factorial moments of the offspring variables ν1,…,νN.

The Kingman coalescent appears in the limit as the population size N tends to infinity if and only if triple mergers are asymptotically negligible in comparison with binary mergers. In this case, a simple upper bound for the rate of convergence of the finite-dimensional distributions is derived. It depends (up to some constants) only on the three factorial moments E((ν1)2), E((ν1)3) and E((ν1)2(ν2)2), where (x)k := x(x-1)…(x-k+1). Examples are the Wright-Fisher model, where the rate of convergence is of order N-1, and the Moran model, with a convergence rate of order N-2.

Let ℬs([a,b];μ1,μ2,…,μs-1) be the class of all distribution functions of random variables with support in [a,b] having μ1,μ2,…,μs-1 as their first s-1 moments. In this paper we examine some aspects of the structure of ℬs([a,b];μ1,μ2,…,μs-1) and of the s-convex stochastic extrema in it. Using representation results of moment matrices à la Lindsay (1989a), we provide conditions for the admissibility of moment sequences in ℬs([a,b];μ1,μ2,…,μs-1) in terms of lower bounds on the number of support points of the corresponding distribution functions. We point out two special distributions with a minimal number of support points that are the s-convex extremal distributions. It is shown that the support points of these extrema are the roots of some polynomials, and an efficient method for the complete determination of the distribution functions of these extrema is described. A study of the goodness of fit, of the approximation of an arbitrary element in ℬs([a,b];μ1,μ2,…,μs-1) by one of the stochastic s-convex extrema, is then given. Using standard ideas from linear regression, we derive Tchebycheff-type inequalities which extend previous results of Lindsay (1989b), and we establish some limit theorems involving the moment matrices. Finally, we describe some applications in insurance theory, namely, we provide bounds on Lundberg's coefficient in risk theory, and on the actual interest rate relating to a life insurance policy. These bounds are obtained with the aid of the s-convex extrema, and are determined only by the support and the first few moments of the underlying distribution.

Suppose X1,X2 are independent random variables satisfying a second-order regular variation condition on the tail-sum and a balance condition on the tails. In this paper we give a description of the asymptotic behaviour as t → ∞ for P(X1 + X2 > t). The result is applied to the problem of risk diversification in portfolio analysis and to the estimation of the parameter in a MA(1) model.

We consider a birth and growth model where points (‘seeds’) arrive on a line randomly in time and space and proceed to ‘cover’ the line by growing at a uniform rate in both directions until an opposing branch is met; points which arrive on covered parts of the line do not contribute to the process. Existing results concerning the number of seeds assume that points arrive according to a Poisson process, homogeneous on the line, but possibly inhomogeneous in time. We derive results under less stringent assumptions, namely that the arrival process be a stationary simple point process.

For a stationary long-range dependent point process N(.) with Palm distribution P0, the Hurst index H ≡ sup{h : lim sup t→∞t-2h var N(0,t] = ∞} is related to the moment index κ ≡ sup{k : E0(Tk) < ∞} of a generic stationary interval T between points (E0 denotes expectation with respect to P0) by 2H + κ ≥ 3, it being known that equality holds for a stationary renewal process. Thus, a stationary point process for which κ < 2 is necessarily long-range dependent with Hurst index greater than ½. An extended example of a Wold process shows that a stationary point process can be both long-range count dependent and long-range interval dependent and have finite mean square interval length, i.e., E0(T2) < ∞.

For continuous-time Markov chains with semigroups P, P' taking values in a partially ordered set, such that P ≤ stP', we show the existence of an order-preserving Markovian coupling and give a way to construct it. From our proof, we also obtain the conditions of Brandt and Last for stochastic domination in terms of the associated intensity matrices. Our result is applied to get necessary and sufficient conditions for the existence of Markovian couplings between two Jackson networks.

This paper introduces a fractional heat equation, where the diffusion operator is the composition of the Bessel and Riesz potentials. Sharp bounds are obtained for the variance of the spatial and temporal increments of the solution. These bounds establish the degree of singularity of the sample paths of the solution. In the case of unbounded spatial domain, a solution is formulated in terms of the Fourier transform of its spatially and temporally homogeneous Green function. The spectral density of the resulting solution is then obtained explicitly. The result implies that the solution of the fractional heat equation may possess spatial long-range dependence asymptotically.

A near-maximum is an observation which falls within a distance a of the maximum observation in an i.i.d. sample of size n. The asymptotic behaviour of the number Kn(a) of near-maxima is known for the cases where the right extremity of the population distribution function is finite, and where it is infinite and the right hand tail is exponentially small, or fatter than exponential. This paper completes the picture for thin tails, i.e., tails which decay faster than exponential. Limit theorems are derived and used to find the large-sample behaviour of the sum of near-maxima.

We study the (two-sided) exit time and position of a random walk outside boundaries which are regularly varying functions of smaller order at infinity than the square root. A natural domain of interest is those random walks which are attracted without centring to a normal law, or are relatively stable. These are shown to have ‘stable’ exit positions, in that the overshoot of the curved boundary is of smaller order of magnitude (in probability) than the boundary, as the boundary expands. Surprisingly, this remains true regardless of the shape of the boundary. Furthermore, within the same natural domain of interest, norming of the exit position by, for example, the square root of the exit time (in the finite-variance case), produces limiting distributions which are computable from corresponding functionals of Brownian motion. We give a functional limit theorem for attraction of normed sums to general infinitely divisible random variables, as a means of making this, and more general, computations. These kinds of theorems have applications in sequential analysis, for example.

We consider a fluid queue fed by the superposition of n homogeneous on-off sources with generally distributed on and off periods. The buffer space B and link rate C are scaled by n, so that we get nb and nc, respectively. Then we let n grow large. In this regime, the overflow probability decays exponentially in the number of sources n. We specifically examine the scenario where b is also large. We obtain explicit asymptotics for the case where the on periods have a subexponential distribution, e.g., Pareto, Lognormal, or Weibull.

The results show a sharp dichotomy in the qualitative behavior, depending on the shape of the function v(t) := - logP(A* > t) for large t, A* representing the residual on period. If v(.) is regularly varying of index 0 (e.g., Pareto, Lognormal), then, during the path to overflow, the input rate will only slightly exceed the link rate. Consequently, the buffer will fill ‘slowly’, and the typical time to overflow will be ‘more than linear’ in the buffer size. In contrast, if v(.) is regularly varying of index strictly between 0 and 1 (e.g., Weibull), then the input rate will significantly exceed the link rate, and the time to overflow is roughly proportional to the buffer size.

In both cases there is a substantial fraction of the sources that remain in the on state during the entire path to overflow, while the others contribute at their mean rates. These observations lead to approximations for the overflow probability. The approximations may be extended to the case of heterogeneous sources. The results provide further insight into the so-called reduced-load approximation.

This paper presents the large-buffer asymptotics for a multiplexer which serves N types of heterogeneous sessions which have long-tailed session lengths. Specifically, the model considered is that sessions of type i ∈ {1,…,N} arrive as a Poisson process with rate λi. Each type of session (independently) remains active for a random duration, say τi, where P(τi > x) ~ αix-(1 + βi) for positive numbers αi and βi. While active, a session transmits at a rate ri. Under the assumption that the average load ρ = ∑Ni=1riλiE[τi] < C, where C denotes the server capacity, we show that both the tail distribution of the stationary buffer content and the loss asymptotics in finite buffers of size z behave approximately as z-κJ0, where κJ0 depends not only on the βi but also on the transmission rates ri; it is the ratio of βi to ri which determines κJ0. When specialized to the homogeneous case, i.e., when ri=r and βi = β for all i, the result coincides with results reported in the literature which have been shown under more restrictive hypotheses. Finally, it is a simple observation that light-tailed sessions only have the effect of reducing the available capacity for long-tailed sessions, but do not contribute otherwise to the definition of κJ0.

The scientific community is saddened to learn of the death of Professor Georges Matheron on 7 August 2000. As a major figure in applied mathematics over four decades, he leaves an outstanding scientific heritage, covering a wide range of domains where probabilistic tools and models are implemented. He also trained more than one generation of researchers, engineers and teachers. In the fields of theoretical and applied random media and image analysis his work has inspired many researchers worldwide: mathematicians, statisticians, physicists, experimentalists, earth scientists, mining engineers and also users of image analysis and synthesis in all its applications (materials, biology, artificial vision, CAD, remote sensing, geophysics, biometry, image coding etc.).